Inscribe a triangle D'E'F' in ABC, similar to DEF and having side D'E' parallel to given direction HG.
The point O, common to the circumcircles of triangles: D'E'B, CD'F', AE'F' remains the same as GH changes direction. (CTRL+2 and pick-move point G or/and point H).
Concerning the story about O look at InscribedTria_In_Tria2.html .